L2 cohomology

In mathematics, L² cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form.

L² cohomology, which grew in part out of L² d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979). It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology.

Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L² cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Looijenga (1988) and by Saper and Stern (1990).

References

• Atiyah, M. F (1976). "Elliptic operators, discrete groups and von Neumann algebras". Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974). Paris: Soc. Math. France. pp. 43–72. Ast\'erisque, No. 32–33.
• B. Brent Gordon (2001), "Baily–Borel compactification", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Cheeger, Jeff Spectral geometry of singular Riemannian spaces. J. Differential Geom. 18 (1983), no. 4, 575–657 (1984).MR 0730920
• Cheeger, Jeff On the Hodge theory of Riemannian pseudomanifolds. Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 91–146, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980. MR 0573430
• Cheeger, Jeff On the spectral geometry of spaces with cone-like singularities. Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 0530173
• J. Cheeger, M. Goresky, R. MacPherson, L² cohomology and intersection homology for singular algebraic varieties, Seminar on differential geometry, vol. 102 of Annals of Mathematics Studies, pages 303–340.MR 0645745
• M. Goresky L² cohomology is intersection cohomology
• Frances Kirwan, Jonathan Woolf An Introduction to Intersection Homology Theory,, chapter 6 ISBN 1-58488-184-4
• Looijenga, Eduard L²-cohomology of locally symmetric varieties. Compositio Mathematica 67 (1988), no. 1, 3–20. MR 0949269
• Lueck, Wolfgang (2002). L²-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 44. Berlin: Springer-Verlag. ISBN 3-540-43566-2.
• Saper, Leslie; Stern, Mark L₂-cohomology of arithmetic varieties. Ann. of Math. (2) 132 (1990), no. 1, 1–69. MR 1059935
• Zucker, Steven, Théorie de Hodge à coefficients dégénérescents. Comptes Rendus Acad. Sci. 286 (1978), 1137–1140.
• Zucker, Steven, Hodge theory with degenerating coefficients: L²-cohomology in the Poincaré metric. Annals of Math. 109 (1979), 415–476.
• Zucker, Steven, L²-cohomology of warped products and arithmetic groups. Inventiones Math. 70 (1982), 169–218.